REVISION
CAMBRIDGE IGCSE 2024
0580
MATHEMATICS
EXTENDED
CHAPTER 4
GEOMETRY AND SYMMETRY
1
(a) Construct DPQR such that PQ = 8.4 cm, PR = 7.2 cm and QR = 9.8 cm.
[3]
(b) Measure and write down the size of the angle facing the longest side.
Answer: ………………………………………… [2]
(c) Construct the angle bisector of Q𝑃"R such that it cuts QR.
Measure and write down the length of QT, such that T is the point where the angle
bisector of Q𝑃"R cuts QR.
Answer: ………………………………………… [3]
2
(a) Construct DABC such that AB = 7.6 cm, BC = 4.8 cm and A𝐵"C = 130°.
[3]
(b) Measure and write down the length of AC.
Answer: ………………………………………… [1]
SIMILAR FIGURES
If two figures are similar, given that a and b are corresponding sides of the two similar figures,
then the following ratios and proportions hold true.
RATIO OF SIDES a:b a:b=c:d
RATIO OF AREAS a2 : b2 a2 : b2 = A1: A2
RATIO OF VOLUME a3 : b3 a3 : b3 = V1: V2
10
Sarah investigates cylindrical plant pots.
The standard pot has base radius r cm and height h cm.
Pot A has radius 3r and height h. Pot B has radius r and height 3h.
Pot C has radius 3r and height 3h.
(a) Write down the volumes of pots A, B and C in terms of 𝜋, r and h.
Answer: ………………………………………… [3]
Answer: ………………………………………… [4]
Answer: ………………………………………… [3]
(b) Find in its lowest terms the ratio of the volumes of A : B : C.
Answer: ………………………………………… [2]
(c) Which one of the pots A, B or C is mathematically similar to the standard pot?
Explain your answer.
Answer: ………………………………………………………………………… [2]
(d) The surface area of the standard pot is S cm2.
Write down the surface area of the similar pot in terms of S.
Answer: ………………………………………… [2]
13.
The two cones are similar.
(a) Write down the value of l.
Answer: ………………………………………… [1]
(b) When full, the larger cone contains 172 cm3 of water.
How much water does the smaller cone contain when it is full?
Answer: ………………………………………… [2]
15 The area of triangle APQ is 99 cm2 and the area of triangle ABC is 11 cm2.
BC is parallel to PQ and the length of PQ is 12 cm.
Calculate the length of BC.
Answer: ………………………………………… [3]
8 A model of a car has a scale of 1 : 25.
(a) The length of the car is 3.95 m.
Calculate the length of the model.
Give your answer in centimetres.
Answer: ………………………………………… [3]
(b) The painted surface area of the model is 128 cm2.
Calculate the painted surface area of the car, giving your answer in square centimetres.
Answer: ………………………………………… [2]
(c) The size of the luggage space of the car is 250 litres.
Calculate the size of the luggage space of the model, giving your answer in millilitres.
Answer: ………………………………………… [3]
13 A statue two metres high has a volume of five cubic metres.
A similar model of the statue has a height of four centimetres.
(a) Calculate the volume of the model statue in cubic centimetres.
Answer: ………………………………………… [2]
(b) Write your answer in part (a) in cubic metres.
Answer: ………………………………………… [1]
2 The volumes of two similar cones are 36𝜋 cm3 and 288 𝜋 cm3.
The base radius of the smaller cone is 3 cm.
Calculate the base radius of the larger cone.
Answer: ………………………………………… [3]
The diagram shows two similar figures.
The areas of the figures are 5 cm2 and 7.2 cm2.
The lengths of the bases are l cm and 6.9 cm.
Calculate l.
Answer: ………………………………………… [3]
4
The two containers are mathematically similar in shape.
The larger container has a volume of 3456 cm3 and a surface area of 1024 cm2.
The smaller container has a volume of 1458 cm3.
Calculate the surface area of the smaller container.
Answer: ………………………………………… [4]
Triangle ABC is similar to triangle DEF.
Calculate the value of
(a) x
Answer: ………………………………………… [2]
(b) y
Answer: ………………………………………… [2]
CONGRUENT FIGURES
(a) Two figures are congruent if all the corresponding parts [sides and angles] are equal.
(b) Two triangles are congruent if at least 3 corresponding parts are equal:
7
The diagram shows two straight lines, AE and BD, intersecting at C.
Angle ABC and angle EDC are equal.
Triangles ABC and EDC are congruent.
Write down two properties of line segments AB and DE.
Answer: AB and DE are …………………………………………
and ………………………………………… [2]
5
These two triangles are congruent.
Write down the value of
(a) x
Answer: ………………………………………… [1]
(b) y
Answer: ………………………………………… [1]
SYMMETRY
LINE OF SYMMETRY ROTATIONAL SYMMETRY
- Same as Line of Reflection - After rotating to a certain angle, the image
- A line that divides an object into two (2) will be EXACTLY the same as the object,
images reflecting each other e.i. position, orientation, color, shape…
PLANE OF SYMMETRY ORDER OF ROTATIONAL SYMMETRY
- A flat surface [like an A4 paper] that - Number [of times] of rotations such that
divides a 3D object into two (2) 3D images IMAGE = OBJECT.
reflecting each other
1 T R I G O N O M E T R Y
From the word above, write down the letters which have
(a) exactly two lines of symmetry.
Answer: ………………………………………… [1]
(b) rotational symmetry of order 2.
Answer: ………………………………………… [1]
8 (a) (i) Complete quadrilateral ABCD so that the dotted line is the only line of symmetry.
[1]
(ii) Write down the special name for quadrilateral ABCD.
Answer: ………………………………………… [1]
(b) (i) Complete quadrilateral EFGH so that the dotted line is one of two lines of
symmetry. [1]
(ii) Write down the special name for quadrilateral EFGH.
Answer: ………………………………………… [1]
1
For the diagram above, write down
(a) the order of rotational symmetry,
Answer: ………………………………………… [1]
(b) the number of lines of symmetry.
Answer: ………………………………………… [1]
For the diagram above, write down
(a) the order of rotational symmetry,
Answer: ………………………………………… [1]
(b) the number of lines of symmetry.
Answer: ………………………………………… [1]
22
Diagram 1 Diagram 2 Diagram 3
(a) Write down the order of rotational symmetry of each diagram.
Answer (Diagram 1): …………………………… [1]
Answer (Diagram 2): …………………………… [1]
Answer (Diagram 3): …………………………… [1]
(b) Draw all the lines of symmetry on each diagram. [3]
Basic Angle Properties
Angles in a Triangle: Angles in a Triangle/Quadrilateral:
The sum of angles in a triangle is 180° The sum of angles in a quadrilateral is 360°.
Angles in a Polygon
Angle Properties of Circles
Angle in a semi-circle is a right angle. Angles in the same segment are equal.
Angle at the centre of the circle is twice the angle Angles in opposite segments are
at the circumference. supplementary/cyclic quadrilaterals.
Angle between the tangent and radius/diameter of Alternate segment theorem
a circle is right angle
Answer: ………………………………………… [3]
Answer: …………………………………………………………………………..…… [1]
Answer: ………………………………………… [1]
Answer: ………………………………………… [1]
